It is not hard to verify that the following non-recursive description of shift
digraphs is equivalent to Definition REF  for \(i\ge 1\) : \(\mathbf {S}_{q,i}\)  is
the digraph whose vertex set consists of all strings of length \(i+1\)  over the
alphabet \(\lbrace 1,\ldots ,q\rbrace \)  such that consecutive letters are distinct, and whose edge set consists of all pairs of strings of the form \((a_1\dots a_k,\;a_2\dots a_{k+1})\) .In [1]}, a slightly different definition of shift digraphs is given, where the case \(i=0\)  is a transitive tournament rather than a clique; equivalently, the vertex set of \(\mathbf {S}_{q,i}\)  only includes monotonically increasing strings.
In particular, it is clear from this description that the edge set of \(\mathbf {S}_{q,i}\)  is nonempty for \(q\ge 2\) .
